number skills - wiley · concept map to show your class’s knowledge of real numbers. ... number...

2.1 OverviewWhy learn this?Mathematics has been a part of every civilisation throughout history. From the most primitive times people have needed numbers for counting and calculating. Our modern world is linked by computers, which rely heavily on numbers to store, fi nd and track information.

What do you know? 1 THInK List what you know about numbers. Use a thinking

tool such as a concept map to show your list.2 PaIr Share what you know with a partner, then with a small group.3 SHare As a class, create a thinking tool such as a large

concept map to show your class’s knowledge of real numbers.

Learning sequence2.1 Overview2.2 Rational numbers2.3 Surds2.4 Real numbers2.5 Scientifi c notation2.6 Review ONLINE ONLY

Number skills

TOPIC 2

number and algebra

c02NumberSkills.indd 12 7/5/14 3:14 PM

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WaTCH THIS vIdeOThe story of mathematics:The evolution of numbers

Searchlight Id: eles-1689


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14 Maths Quest 9

number and algebra

2.2 Rational numbersNatural numbers • Numbers were first used for counting. • The numbers 1, 2, 3, 4 . . . are called natural numbers, and the set of natural numbers is

called N. That is, positive whole numbers are called natural numbers (N). • Natural numbers can be used to answer questions such as:

5 + 43 = ?8 × 6 = ?

Integers • What is the value of 3 − 8?

The answer to this question is not a natural number, so zero and the negative numbers were defined and a name was given to this new list of numbers.

• The numbers . . . −3, −2, −1, 0, 1, 2, 3 . . . are called whole numbers or integers, and the set of integers is called Z.

• Can you suggest any reasons why we need negative numbers?

Rational numbers • What is the value of 4 ÷ 7?

The answer to this question is not an integer, but a fraction (or ratio) written as 47.

• Fractions such as 47 or −11

3 in which the denominator and numerator are both integers are

called rational numbers. The set of rational numbers is called Q. • Can you suggest any reasons why we need fractions?


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Topic 2 • Number skills 15

• Note: The quotient 9 ÷ 0 has no answer; therefore, numbers such as 90 do not exist. They

are said to be ‘undefi ned’. Because they ‘do not exist’, they are not rational numbers.

By writing each of the following in fraction form, show that the numbers are rational.a 7 b −11 c 0 d −42

5 e 1.2

THInK WrITe

Each number must be written as a fraction using integers.

a The number 7 has to be written in fraction form. To write a number as a fraction, it must be written with a numerator and denominator. For a whole number, the denominator is 1.

a 7 = 71

is rational.

b The number −11 has to be written in fraction form. To write a number as a fraction, it must be written with a numerator and denominator. For a whole number, the denominator is 1.

b −11 = −111

is rational.

c The number 0 has to be written in fraction form. To write a number as a fraction, it must be written with a numerator and denominator. For a whole number, the denominator is 1.

c 0 = 01

is rational.

d Change −425 to an improper fraction. d −42

5= −22

5 is rational.

WOrKed eXamPle 1WOrKed eXamPle 1WOrKed eXamPle 1


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16 Maths Quest 9

e The number 1.2 can be expressed as 1 + 0.2. This can then be expressed as

1 + 210

. Write this as an improper fraction.

e 1.2 = 1210

is rational.

Rational numbers written as decimals • When a rational number is written as a decimal there are two possibilities.

1. The decimal terminates, e.g. 54

= 1.25.

2. The decimal repeats, or recurs, e.g. 76

= 1.1666 . . .

For 1.1666 . . . the 6 is a repeating digit. This number, 76, is called a recurring decimal,

and it can also be written as 1.16#.

1.6#15

# or 1.615 means 1.615 615 615 . . . (The digits 615 in the decimal repeat.)

1.61#5# or 1.615 means 1.615 151 515 . . . (The digits 15 in the decimal repeat.)

1.615# means 1.615 555 . . . (Only the digit 5 in the decimal repeats.)

Write the fi rst 8 digits of each of the following recurring decimals.a 3.0

#2# b 47.1

# c 11.54

#9#

THInK WrITe

a In 3.0#2#, 02 recurs. a 3.020 202 0

b In 47.1#, 1 recurs. b 47.111 111

c In 11.54#9#, 49 recurs. c 11.549 494

Write each fraction as a recurring decimal using dot notation. (Use a calculator for the division.)

a 56 b 57

99 c 25

11 d 4

7

THInK WrITe

a 5 ÷ 6 = 0.833 333 33 recurs — the dot goes above the 3.

a56

= 0.83#

b 57 ÷ 99 = 0.575 757 575 757 recurs — dots go above the 5 and the 7.

b5799

= 0.5#7#

c 25 ÷ 11 = 2.272 727 2727 recurs — dots go above the 2 and the 7.

c2511

= 2.2#7#

d 4 ÷ 7 = 0.571 428 571It looks as though 571 428 will recur — dots go above the 5 and the 8.

d47

= 0.5#71428

#




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Is every decimal a rational number? • 1.237 is rational, because it is a terminating decimal. It is easy to show that

1.237 = 12371000

.

• 0.8#6#, or 0.868 686 . . ., is rational because it is a recurring decimal. It can be shown that

0.8#6#

= 8699

.

• Decimals that do not terminate and do not recur are not rational. They cannot be written as integer fractions or ratios, and are called irrational numbers.

The number system

Rational numbers Q

IntegersZ

Non-integer rationals(terminating and

recurring decimals)

Irrational numbers I(surds, non-terminating

and non-recurringdecimals, π, e)

Negative integersZ‒

Positive integersZ+

(naturalnumbers N )

Zero(neither positive

nor negative)

Real numbers R

Note: A real number is any number that lies on the number line. Further explanation can be found in the next section.

Exercise 2.2 Rational numbers IndIvIdual PaTHWaYS

⬛ PraCTISeQuestions:1a–l, 2a–f, 3a–e, 4–7

⬛ COnSOlIdaTeQuestions:1d–l, 2c–j, 3c–g, 4–11

⬛ maSTerQuestions:1e–l, 2f–j, 3e–j, 4–11

reFleCTIOn How many recurring digits will

there be for 1

13?

⬛ ⬛ ⬛ Individual pathway interactivity int-####


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18 Maths Quest 9

FluenCY

1 WE1 Show that the following numbers are rational by writing each of them in fraction form.

a 15 b −8 c 223 d −51

8

e !16 f 7 310

g 0.002 h 87.2

i 0 j 1.56 k 3.612 l −0.08

2 WE2 Write the fi rst 8 digits of each of the following recurring decimals.a 0.5

# b 0.5

#1# c 0.51

# d 6.03

#1# e 5.1

#83

#

f −7.024# g 8.9

#124

# h 5.123

#4# i 5.1

#234

# j 3.00

#2#

3 WE3 Write the following fractions as recurring decimals using dot notation. (Use a calculator.)a

59 b

311

c 2 211

d 37 e

−17399

f 73

990 g −35

6 h

715

i 4699

j 46

990

underSTandIng

4 From the following list of numbers:−3, −3

7, 0, 2.3, 2.3

#, 23

5, 15

a write down the natural numbersb write down the integersc write down the rational numbers.

5 Write these numbers in order from smallest to largest.2.1

#, 2.1

#2#, 2.1

#21

#, 2.12

#1#, 2.12

#

reaSOnIng

6 Explain why all integers are rational numbers. 7 Are all fractions in which both the numerator and denominator are integers rational?

Explain why or why not.

PrOblem SOlvIng

8 a Using the fraction ab

, where a and b are natural numbers, write 3 recurring decimals

in fractional form using the smallest natural numbers possible.b What was the largest natural number you used?

9 Write 2 fractions that have the following number of repeating digits in their decimal forms.a 1 repeating digitb 2 repeating digitsc 3 repeating digitsd 4 repeating digits

10 If a $2 coin weighs 6 g, a $1 coin weighs 9 g, a 50c coin weighs 15 g, a 20c coin weighs 12 g, a 10c coin weighs 5 g and a 5c coin weighs 3 g, explain what the maximum value of the coins would be if a bundle of them weighed 10 kg.

11 A Year 9 student consumes 2 bottles of water every day, except on every 5th day when she only has 1. Calculate her annual bottled water consumption.

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2.3 Surds • Ancient mathematicians were shocked to fi nd that rational numbers could not be used to

label every point on the number line. In other words they discovered lengths that could not be expressed as fractions. These numbers are called irrational numbers.

• A number is irrational if it is not rational; that is, if it cannot be written as a fraction, nor as a terminating or recurring decimal.

• Irrational numbers are denoted by the letter I.

Roots • The nth root of any positive number can be found. That is, "n

b = a or an = b. For example, !25 = 5 because 25 = 5 × 5. Note: The 2 is usually not written in square roots, for example !25 = "2 25. "3 64 = 4 because 64 = 4 × 4 × 4.

Evaluate each of the following.a !16 b "3 27 c "5 32

THInK WrITe

a Determine the number that when multiplied by itself gives 16 : 16 = 4 × 4.

a !16 = 4

b Determine the number that when multiplied by itself three times gives 27 : 27 = 3 × 3 × 3.

b "3 27 = 3

c Determine the number that when multiplied by itself fi ve times gives 32: 32 = 2 × 2 × 2 × 2 × 2.

c "5 32 = 2

int-2762int-2762int-2762



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20 Maths Quest 9

Surds • When the square root of a number is an irrational number, it is called a surd. For

example, !10 cannot be written as a fraction, nor as a recurring or terminating decimal. It is therefore irrational and is called a surd.

!10 ≈ 3.162 277 660 17. . . • The value of a surd can be approximated using a number line.

For example, !21 will lie between 4 and 5, because !16 = 4 and !25 = 5. This can be shown on a number line.

3 4 5 6 7

16 25

21(approximately)

Place !34 on a number line.

THInK WrITe

1 The square number that is smaller than 34 is 25. The square number that is larger than 34 is 36.Write the numbers just below and just above !34.

!34 lies between !25 and !36.

2 Draw a number line to show the approximate position of !34. 4 5 6 7

25 36

34

• Note: Negative numbers do not have square roots within the set of real numbers. However, in the 18th century mathematicians described the square root of a negative number as an ‘imaginary number’. Imaginary numbers and real numbers together form the set of complex numbers. Imaginary numbers have many uses in higher level mathematics used in science and engineering, but they are beyond the scope of this current course.

Which of the following are surds?a !0 b !20 c −!9 d "3 6

THInK WrITe

a !0 = 0. This is a rational number and therefore not a surd.

a !0 = 0, which is not a surd.

b !20 ≈ 4.472 135 955 . . . It is an irrational number and therefore a surd.

b !20 is a surd.

c −!9 = −3. This is a rational number and therefore not a surd.

c −!9 = −3, which is not a surd.

d "3 6 ≈ 1.817 120 592 83 . . . It is an irrational number and therefore a surd.

d "3 6 is a surd.




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Multiplying and dividing surds • Consider that 3 = !9, 2 = !4, and 6 = !36.

The multiplication 3 × 2 = 6 could be written as !9 × !4 = !36.

• In general, !a × !b = !ab.For example, !7 × !3 = !21.

• Similarly, !a ÷ !b = Äab

.

For example, !18 ÷ !3 = !6.

Evaluate the following, leaving your answer in surd form.

a !7 × !2 b 5 × !3 c !5 × !5 d −2!3 × 4!5

THInK WrITe

a Apply the rule !a × !b = !ab. a !7 × !2 = !14

b Only !3 is a surd. It is multiplied by 5, which is not a surd.

b 5 × !3 = 5!3

c Apply the rule !a × !b = !ab. c !5 × !5 = !25= 5

d Multiply the whole numbers by each other. Multiply the surds by each other.

d −2!3 × 4!5 = −2 × 4 × !3 × !5= −8 × !15= −8!15

Evaluate the following, leaving your answer in surd form.

a !10

!5 b Å

105

c −6!8

4!4 d

!20

!5 e

5!5

THInK WrITe

a Apply the rule !a ÷ !b = Äab

. a !10

!5 = Å

105

= !2

b Simplify the fraction fi rst. b Å105

= !2

c Simplify the whole numbers. Then

apply the rule !a ÷ !b = Äab

.

c−6!8

4!4 = −3!2

2




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22 Maths Quest 9

d Apply the rule !a ÷ !b = Äab

. d!20

!5 = !4

= 2

e Rewrite the numerator as the product of two surds and then simplify.

e 5

!5 = !5 × !5

!5

= !5

Simplifying surds • Just as a rational number can be written many different ways (e.g. 1

2= 5

10= 7

14), so can a

surd, and it is expected that surds should normally be written in simplest form. • A surd is in simplest form when the number inside the radical sign has the smallest

possible value. • Note that !24 can be factorised several ways. For example:!24 = !2 × !12!24 = !3 × !8!24 = !4 × !6In the last case !4 = 2:!24 = 2 × !6 = 2!6

2!6 is equal to !24, and 2!6 is written in simplest form.

• To simplify a surd you must fi nd a factor that is also a perfect square, for example 4, 9, 16, 25, 36 and 49.

Simplify the following surds.

a !18 b 6!20THInK WrITe

a 1 Rewrite 18 as the product of two numbers, one of which is square.

a !18 = !9 × !2

2 Simplify. = 3 × !2 = 3!2

b 1 Rewrite 20 as the product of two numbers, one of which is square.

b 6!20 = 6 × !4 × !5

2 Simplify. = 6 × 2 × !5 = 12!5

• The surd !22 cannot be simplifi ed because 22 has no perfect square factors. • Surds can be simplifi ed in more than one step.

!72 = !4 × !18= 2!18

= 2 × !9 × !2 = 2 × 3!2 = 6!2



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Entire surds • The surd !45, when simplifi ed, is written as 3!5.

The surd !45 is called an entire surd, because it is written entirely inside the radical sign, whereas 3!5 is not.

• Writing a surd as an entire surd reverses the process of simplifi cation.

Write 3!7 as an entire surd.

THInK WrITe

1 In order to place the 3 inside the radical sign it has to be written as !9.

3!7 = !9 × !7

2 Apply the rule !a × !b = !ab. = !63

Which number is larger, 3!5 or 5!3?

THInK WrITe

1 Write 3!5 as an entire surd. 3!5 = !9 × !5 = !45

2 Write 5!3 as an entire surd. 5!3 = !25 × !3 = !75

3 Compare the values. !75 > !45

4 Write your answer. 5!3 is the larger number.

Addition and subtraction of surds • Surds can be added or subtracted if they are like terms.

Simplify each of the following.a 6!3 + 2!3 + 4!5 − 5!5 b 3!2 − 5 + 4!2 + 9

THInK WrITe

a Collect the !3’s and the !5’s. a 6!3 + 2!3 + 4!5 − 5!5= 8!3 − !5

b Collect the like terms and simplify. b 3!2 − 5 + 4!2 + 9= 3!2 + 4!2 − 5 + 9= 7!2 + 4

• Surds should be simplifi ed before adding or subtracting like terms.





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24 Maths Quest 9

Simplify 5!75 − 6!12 + !8 − 4!3.

THInK WrITe

1 Simplify 5!75. 5!75 = 5 × !25 × !3 = 5 × 5 × !3 = 25!3

2 Simplify 6!12. 6!12 = 6 × !4 × !3 = 6 × 2 × !3 = 12!3

3 Simplify !8. !8 = !4 × !2 = 2!2

4 Rewrite the original expression and simplify by adding like terms.

5!75 − 6!12 + !8 − 4!3= 25!3 − 12!3 + 2!2 − 4!3= 9!3 + 2!2

Exercise 2.3 Surds IndIvIdual PaTHWaYS

⬛ PraCTISeQuestions:1–9, 11a–f, 12a–f, 14

⬛ COnSOlIdaTeQuestions:1–10, 11e–j, 12e–h, 13–15

⬛ maSTerQuestions:1k–n, 2e–l, 3i–p, 4m–p, 5l–p, 6–10, 11g–l, 12e–j, 13, 15–21

FluenCY

1 WE4 Write down the square roots of each of the following.a 1 b 4 c 0 d

19 e 1 9

16

f 0.16 g 400 h 10 000 i 4

25 j 1.44

k 20.25 l 1 000 000 m 0.0009 n 2562 WE5 Write down the value of each of the following.

a !81 b −!81 c !121 d −!441

e "3 8 f "3 64 g "3 343 h "4 81

i "5 1024 j "3 125 k −!49 l "3 −273 WE6 Which of the following are surds?

a "3 0 b !10 c −!36 d "3 9e −"3 216 f "3 −216 g −!2 h 1 + !2i 1.32 j 1.3

#2# k "4 64 l 1.752 16

m !7 + !2 n !9 + !49 o !4 − !3 p !32

4 WE7 Simplify each of the following.a !3 × !7 b −!3 × !7 c 2 × !6d 2 × 3!7 e 2!7 × 5!2 f −3!2 × −5!5g 3!7 × 4 h !7 × 9 i −2!5 × − 11!2


reFleCTIOnAre all square root numbers surds?


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j 2!3 × 11 k !3 × !3 l 2!3 × 4!3m !6 × !6 n !11 × !11 o !51 × !51p !15 × 2!15

5 WE8 Simplify each of the following.

a Å124

b −Å105

c !18

!3 d

−!15

−!3

e 15!6

5!2 f

15!610

g 15!6

!3 h

53Å

153

i −10!10

5!2 j

!9

!3 k

3

!3 l

7

!7

m 35

!35 n

!28

!7 o

−!18

!2 p

!45

!5 6 WE9 Simplify each of the following.

a !20 b !8 c !18 d !49e !30 f !50 g !28 h !108i !288 j !48 k !500 l !162

7 WE9 Simplify each of the following.a 2!8 b 5!27 c 6!64 d 7!50e 10!24 f 5!12 g 4!42 h 12!72i 9!45 j 12!242

8 WE10 Write each of the following in the form !a; that is, as an entire surd.a 2!3 b 5!7 c 6!3 d 4!5e 8!6 f 3!10 g 4!2 h 12!5i 10!6 j 13!2

9 MC a !1000 is equal to:a 31.6228 b 50!2 C 50!10 d 10!10

b !80 in simplest form is equal to:a 4!5 b 2!20 C 8!10 d 5!16

c Which of the following surds is in simplest form?a !60 b !147 C !105 d !117

d Which of the following surds is not in simplest form?a !102 b !110 C !116 d !118

e 6!5 is equal to:a !900 b !30 C !150 d !180

f Which one of the following is not equal to the rest?a !128 b 2!32 C 8!2 d 64!2

g Which one of the following is not equal to the rest?a 4!4 b 2!16 C 8 d 16

h 5!48 is equal to:a 80!3 b 20!3 C 9!3 d 21!3

10 Challenge: Reduce each of the following to simplest form.a !675 b !1805 c !1792 d !578e "a2c f "bd4 g "h2jk2 h "f

3


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26 Maths Quest 9

11 WE12 Simplify each of the following.a 6!2 + 3!2 − 7!2 b 4!5 − 6!5 − 2!5c −3!3 − 7!3 + 4!3 d −9!6 + 6!6 + 3!6e 10!11 − 6!11 + !11 f !7 + !7g 4!2 + 6!2 + 5!3 + 2!3 h 10!5 − 2!5 + 8!6 − 7!6i 5!10 + 2!3 + 3!10 + 5!3 j 12!2 − 3!5 + 4!2 − 8!5k 6!6 + !2 − 4!6 − !2 l 16!5 + 8 + 7 − 11!5

12 WE13 Simplify each of the following.a !8 + !18 − !32 b !45 − !80 + !5c −!12 + !75 − !192 d !7 + !28 − !343e !24 + !180 + !54 f !12 + !20 − !125g 2!24 + 3!20 − 7!8 h 3!45 + 2!12 + 5!80 + 3!108i 6!44 + 4!120 − !99 − 3!270 j 2!32 − 5!45 − 4!180 + 10!8

13 MC a !2 + 6!3 − 5!2 − 4!3 is equal to:a −5!2 + 2!3 b −3!2 + 23C 6!2 + 2!3 d −4!2 + 2!3

b 6 − 5!6 + 4!6 − 8 is equal to:a −2 − !6 b 14 − !6C −2 + !6 d −2 − 9!6

c 4!8 − 6!12 − 7!18 + 2!27 is equal to:a −7!5 b 29!2 − 18!3C −13!2 − 6!3 d −13!2 + 6!3

d 2!20 + 5!24 − !54 + 5!45 is equal to:a 19!5 + 7!6 b 9!5 − 7!6C −11!5 + 7!6 d −11!5 − 7!6

underSTandIng

14 WE11 Which number is larger?a !10 or 2!3 b 3!5 or 5!2c 10!2 or 4!5 d 2!10 or !20

15 Write these numbers from smallest to largest.

a 6!2, 8, 2!7, 3!6, 4!2, !60

b !6, 2!2, !2, 3, !3, 2, 2!3

reaSOnIng

16 A man wants to divide his vegetable garden into 10 equal squares, each of area exactly 2 m2.a What will be the exact side length of each square? Explain your reasoning.b What will the side length of each square be, correct to 2 decimal places?c The man wishes to group the squares next to each other so as to have the smallest

perimeter for his vegetable garden.i How should he arrange the squares?ii Explain why the exact perimeter of the vegetable patch is 14!2 m.iii What will be the perimeter correct to 2 decimal places?

17 Explain why "a3b2 can be simplified to ab!a.


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PrOblem SOlvIng

18 Kyle wanted a basketball court in his backyard; however, he could not fi t a full-size court in his yard. He did get a rectangular court laid with a width of 6!2 m and length of 3!10 m. Calculate the area of the basketball court and represent it in its simplest surd form.

19 A netball team went on a pre-season training run. They completed 10 laps of a triangular course with side lengths (200!3 + 50) m, (50!2 + 75!3) m and (125!2 − 18) m. What was the distance they ran in simplest surd form?

20 The area of a square is x cm2. Would the side length of the square be a rational number? Explain your answer.

21 To calculate the length of the hypotenuse of a right-angled triangle, use the formula c2 = a2 + b2.a Calculate the length of the hypotenuse for triangles with

other side lengths of:i !5, !8ii !2, !7iii !15, !23

b Describe any pattern in your answers to part a.c State the length of the hypotenuse (without calculations) for triangles with side

lengths of:i !1000, !500ii !423, !18iii !124, !63

d Your friend wrote down the following explanation.!b + !c = !a

Explain why this is not correct and where your friend made an error.

Hypotenusec

a

b

c2 = a2 + b2

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28 Maths Quest 9

2.4 Real numbers • A real number is any number that lies on

the number line. • Together, the rational numbers and the

irrational numbers make up the set of real numbers.

Therefore every point on the number line represents some number, rational or irrational.

The irrational number π • Surds are not the only irrational numbers.

Most irrational numbers cannot be written as surds. A few of these numbers are so important to mathematics and science that they are given special names.

• The most well known of these numbers is π (pi).

• π is irrational, therefore it cannot be written as a fraction. If you tried to write π as a decimal, you would be writing forever, as the digits never recur and the decimal does not terminate. In the 20th century, computers were used to find the value of π to 1 trillion decimal places.

• The value of π is very close to (but not equal to) 3.14 or 227

. Most calculators store an approximate value for π.

–3 –2

–1.625 –0.31

–1 0 1

π√⏤2

2 3 4

52

Rounding decimals • Because numbers such as !3 or π cannot be written exactly as decimals, approximate

values are often used. These values can be found using a calculator and then rounded off to the desired level of accuracy.

• When we round numbers, we write them to a certain number of decimal places or significant figures.

Rounding to decimal places • The rules for rounding to a certain number of decimal places are as follows.

For example, let us round 3.456 734 correct to 3 decimal places. – Identify the rounding digit in the third decimal place. In this case it is 6.

3.456734

– Look at the digit that comes after the rounding digit.

3.456734

If this digit is 0, 1, 2, 3 or 4, leave the rounding digit as it is. If this digit is 5, 6, 7, 8 or 9, then increase the rounding digit by 1. In this case the next digit is 7, so the rounding digit is increased by 1, from 6 to 7.


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– Leave out all digits that come after the rounding digit. So 3.456 734 ≈ 3.457 correct to 3 decimal places (3 d.p.).

Write these numbers correct to 3 decimal places (3 d.p.).a !3 b π c 5.19

# d

23 e 7.123 456

THInK WrITe

a "3 = 1.7320. . . The rounding digit is 2. The next digit is 0, so leave 2 as is.

a !3 ≈ 1.73

b π = 3.1415. . . The rounding digit is 1. The next digit is 5, so add 1.

b π ≈ 3.142

c 5.19# = 5.1999. . . The rounding digit is 9. The next

digit is 9, so add 1.c 5.19

# ≈ 5.200

d23 = 0.6666. . . The rounding digit is 6. The next digit is

6, so add 1.d

23 ≈ 0.667

e 7.123 456. The rounding digit is 3. The next digit is 4, so leave 3 as is.

e 7.1234 ≈ 7.123

Rounding to signifi cant fi gures • Another method of rounding decimals is to write them correct to a certain number of

signifi cant fi gures. In a decimal number the signifi cant fi gures start with the fi rst non-zero digit.

• Consider the approximate value of !2. !2 ≈ 1.414. This approximation is written correct to 3 decimal places, but it has

4 signifi cant fi gures. When counting signifi cant fi gures, the non-zero digits before the decimal point are included.

• The decimal 0.0302 is written to 4 decimal places, but it has only 3 signifi cant fi gures. This is because signifi cant fi gures start with the fi rst non-zero digit. In this case the fi rst signifi cant fi gure is 3, and the signifi cant fi gures are 302.

How many signifi cant fi gures are there in each of the following numbers?

a 25 b 0.04 c 3.02 d 0.100

THInK WrITe

a 25: The fi rst signifi cant fi gure is 2. a 2 signifi cant fi gures

b 0.04: The fi rst signifi cant fi gure is 4. b 1 signifi cant fi gure

c 3.02: The fi rst signifi cant fi gure is 3. c 3 signifi cant fi gures

d 0.100: The fi rst signifi cant fi gure is 1. d 3 signifi cant fi gures




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30 Maths Quest 9

Round these numbers correct to 5 signifi cant fi gures.a π b !200 c 0.0

#3# d 2530.166

THInK WrITe

a π ≈ 3.141 59 . . . The fi rst signifi cant fi gure is 3. Write 4 more digits.

a 3.1416

b !200 ≈ 14.1421 . . . The fi rst signifi cant fi gure is 1. Write 4 more digits.

b 14.142

c 0.0#3# = 0.030 303 0 . . . The fi rst signifi cant

fi gure is 3. Write 4 more digits.c 0.030 303

d 2530.16 . . . The fi rst signifi cant fi gure is 2. Write 4 more digits.

d 2530.2

Exercise 2.4 Real numbers IndIvIdual PaTHWaYS

⬛ PraCTISeQuestions:1–3, 4a–k, 5–11

⬛ COnSOlIdaTeQuestions:1–3, 4e–p, 5–13

⬛ maSTerQuestions:1–3, 4i–t, 5–13

1 WE14 Write each of the following correct to 3 decimal places.a

π2

b !5 c !15 d 5.12 × 3.21

e 5.1# f 5.15

# g 5.1

#5 h e

i 11.722 j 37 k

113

l 2 37

m 0.999 999 n 6.581 29 o 4.000 01 p 2.79 ÷ 11q 0.0254 r 0.000 913 6 s 5.000 01 t 2342.156

2 Write the value of π correct to 4, 5, 6 and 7 decimal places. 3 WE15 How many signifi cant fi gures are there in each of the following numbers?

a 36 b 207 c 1631 d 5.04

e 176.2 f 95.00 g 0.21 h 0.01

i 0.000 316 j 0.1007 k 0.010 l 0.0512 4 WE16 Write each number correct to 5 signifi cant fi gures.

a π2

b !5 c !15 d 5.12 × 3.21

e 5.1# f 5.15

# g 5.1

#5# h e

i 11.722 j 37 k

113

l 237

m 0.999 999 n 6.581 29 o 4.000 01 p 2.79 ÷ 11q 0.0254 r 0.000 913 6 s 5.000 01 t 2342.156

5 List the following numbers in order from smallest to largest.

a !7, 3.5, !18, 4, !15 b 3.14, π, 227

, !10, 3.1#4#


reFleCTIOnIs there an equal number of rational numbers and irrational numbers?


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6 Explain whether each statement is true or false.a Every surd is a rational number. b Every surd is an irrational number.c Every irrational number is a surd. d Every surd is a real number.

7 Explain whether each statement is true or false.a π is a rational number. b π is an irrational number.c π is a surd. d π is a real number.

8 Explain whether each statement is true or false.a 1.3

#1# is a rational number. b 1.3

#1# is an irrational number.

c 1.3#1# is a surd. d 1.3

#1# is a real number.

reaSOnIng

9 Explain why !25 and −!25 are defined, but !−25 is not. 10 Explain why the number 12.995 412 3 when rounded to 2 decimal places

becomes 13.00.

PrOblem SOlvIng

11 The area of a circle is calculated using the formula A = π × r2, where r is the radius of the circle. Pi (π ) is sometimes rounded to 2 decimal places (3.14). A particular circle has a radius of 7 cm.a Use π = 3.14 to calculate the area of the circle to 2 decimal places.b Use the π key on your calculator to calculate the area of the circle to 4 decimal

places.c Round your answer for part b to 2 decimal places.d Are your answers for parts a and c different? Why or why not?

12 The volume of a sphere (a ball shape) is calculated using the formula V = 4

3× π × r3, where r is the radius of the sphere. A

beach ball with a radius of 25 cm is bouncing around the crowd at the MCG during the Boxing Day Test.a Calculate the volume of the beach ball to 4 decimal places.b When the volume is calculated to 4 decimal places, how

many significant figures does it have?c Is the calculated volume a rational number? Why or why not?

13 In a large sample of written English prose there are about 7 vowels to every 11 consonants. The letter e accounts for about one-third of the occurrence of vowels. How many times would you expect the letter e to occur in a passage of 100 000 letters? Round your answer to the nearest 100.

2.5 Scientific notation • Did you know that Jupiter is approximately

778 547 200 km from the sun? Astronomers and other scientists frequently use large numbers that are difficult to read and manipulate. To make them easier to use and compare, they can be written in a special way called scientific notation (or standard form).


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32 Maths Quest 9

• A number written in scientifi c notation looks like this:

5.316 × 102

A number between × A power of 101 and 10

Here are some numbers written in scientifi c notation. Write them in decimal notation.a 7.136 × 102 b 5.017 × 105 c 8 × 106

THInK WrITe

a Move the decimal point 2 places to the right, since 102 = 100.

a 7.136 × 102 = 713.6

b Move the decimal point 5 places to the right, since 105 = 100 000.

b 5.017 × 105 = 501 700

c Move the decimal point 6 places to the right, since 106 = 1 000 000.

c 8 × 106 = 8 000 000

Using a calculator • Numbers written in scientifi c notation can be entered into your calculator using a special

button. Find out how this works on your calculator. • Some calculators have unusual ways of writing in scientifi c notation. For example, some

show 5.71 × 104 as 5.71E4. In writing you should always show this as 5.71 × 104.

a Write each of these numbers in scientifi c notation.b Round each answer to 3 signifi cant fi gures. i 827.2 ii 53 681 iii 51 900 000 000

THInK WrITe

a i The fi rst part needs to be a number between 1 and 10, so 8.272. The decimal point must be moved 2 places (× 102).

a i 827.2 = 8.272 × 102

ii The fi rst part needs to be a number between 1 and 10, so 5.3681. The decimal point must be moved 4 places (× 104).

ii 53 681 = 5.3681 × 104

iii The fi rst part needs to be a number between 1 and 10, so 5.190. The decimal point must be moved 10 places (× 1010).

iii 51 900 000 000 = 5.190 × 1010

b i 8.272 ≈ 8.27 b i 8.27 × 102

ii 5.368 ≈ 5.37 ii 5.37 × 104

iii 5.190 ≈ 5.19 iii 5.19 × 1010




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Small numbers in scientifi c notation • Very small numbers, such as the length or the mass of a

molecule, can also be expressed in scientifi c notation.0.0412 = 4.12 ÷ 102

= 4.12 × 10−2

It will be shown later that 10−2 = 1102

.

• Just as6.285 × 102 = 628.5(the decimal point is moved 2 places to the right),6.285 × 10−2 = 0.062 85(the decimal point is moved 2 places to the left).Similarly,6.285 × 10−1 = 0.6285and6.285 × 10−3 = 0.006 285.

Write these numbers in decimal notation.a 9.12 × 10−1 b 7.385 × 10−2 c 6.32 × 10−7

THInK WrITe

a Move the decimal point 1 place to the left. a 9.12 × 10−1 = 0.912

b Move the decimal point 2 places to the left. (Insert a zero.)

b 7.385 × 10−2 = 0.073 85

c Move the decimal point 7 places to the left. (Insert zeros.)

c 6.32 × 10−7 = 0.000 000 632

Write these numbers in scientifi c notation.a 0.0051 b 0.1321 c 0.000 000 000 7

THInK WrITe

a The fi rst part needs to be a number between 1 and 10, so 5.1. The decimal point must be moved 3 places (× 10−3).

a 0.0051 = 5.1 × 10−3

b The fi rst part needs to be a number between 1 and 10, so 1.321. The decimal point must be moved 1 place (× 10−1).

b 0.1321 = 1.321 × 10−1

c The fi rst part needs to be a number between 1 and 10, so 7.0. The decimal point must be moved 10 places (× 10−10).

c 0.000 000 000 7 = 7.0 × 10−10




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number and algebra

34 Maths Quest 9

Exercise 2.5 Scientifi c notation IndIvIdual PaTHWaYS

⬛ PraCTISeQuestions:1–7, 9, 11, 13–15

⬛ COnSOlIdaTeQuestions:1–8, 10, 12–15, 18

⬛ maSTerQuestions:1–19

FluenCY

1 WE17 Write these numbers in decimal notation.a 6.14 × 102 b 6.14 × 103 c 6.14 × 104 d 3.518 × 102

e 1 × 109 f 3.926 73 × 102 g 5.911 × 102 h 5.1 × 103

i 7.34 × 105 j 7.1414 × 106 k 3.51 × 10 l 8.05 × 104

2 WE18a Write these numbers in scientifi c notation.

a 5000 b 431 c 38 d 350 000

e 72.5 f 725 g 7250 h 725 000 000

3 WE18b Write these numbers in scientifi c notation, correct to 4 signifi cant fi gures.

a 43.792 b 5317 c 258.95 d 110.11

e 1 632 000 f 1 million g 123 456 789 h 249.9

4 WE19 Write these numbers in decimal notation.

a 2 × 10−1 b 4 × 10−3 c 7 × 10−4 d 3 × 10−2

e 8.273 × 10−2 f 7.295 × 10−2 g 2.9142 × 10−3 h 3.753 × 10−5

i 5.29 × 10−4 j 3.3333 × 10−5 k 2.625 × 10−9 l 1.273 × 10−15

5 WE20 Write these numbers in scientifi c notation.

a 0.7 b 0.005 c 0.000 000 3 d 0.000 000 000 01

e 0.231 f 0.003 62 g 0.000 731 h 0.063

6 Write these numbers in scientifi c notation, correct to 3 signifi cant fi gures.

a 0.006 731 b 0.142 57 c 0.000 068 3 d 0.000 000 005 12

e 0.0509 f 0.012 46 g 0.000 731 h 0.063

underSTandIng

7 Write these numbers in ascending order.

a 8.31 × 102, 3.27 × 103, 9.718 × 102, 5.27 × 102

b 7.95 × 102, 4.09 × 102, 7.943 × 102, 4.37 × 102

c 5.31 × 10−2, 9.29 × 10−3, 5.251 × 10−2, 5.27 × 10−1

d 8.31 × 102, 3.27 × 103, 7.13 × 10−2, 2.7 × 10−3

8 One carbon atom weighs 1.994 × 10−23 g.

a Write the weight as a decimal.

b How much will 1 million carbon atoms weigh?

c How many carbon atoms are there in 10 g of carbon? Give your answer correct to 4 signifi cant fi gures.

reFleCTIOn What is the advantage of converting numbers into scientifi c notation?


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9 The distance from Earth to the Moon is approximately 3.844 × 105 km. If you could drive there without breaking the speed limit (100 km/h), how long would it take? What is this in days (correct to 2 decimal places)?

10 The Earth weighs 5.97 × 1024 kg, but the Sun weighs 1.99 × 1030 kg. How many Earths would it take to balance the Sun’s weight? Give your answer correct to 2 decimal places.

11 Inside the nucleus of an atom, a proton weighs 1.6726 × 10−27 kg and a neutron weighs 1.6749 × 10−27 kg. Which one is heavier and by how much?

12 The Earth’s orbit has a radius of 7.5 × 107 km; the orbit of Venus is equal to 5.4 × 107 km. How far apart are the planets when:a they are closest to each other?b they are farthest apart from each other?

reaSOnIng

13 a State the power(s) of 10 that you believe equations i and ii will have when solved. i 5.36 × 107 + 2.95 × 103 ii 5.36 × 107 − 2.95 × 103

b Evaluate equations i and ii, correct to 3 significant figures.c Were your answers to part a correct? Why or why not?

14 Explain why 2.39 × 10−3 + 8.75 × 10−7 = 2.39 × 10−3, correct to 3 significant figures.

PrOblem SOlvIng

15 Write the following numbers in ascending order. 14%, 0.753, 5

8, !2, 0.52, "3 2, 1.5 × 10−2

16 Distance is equal to speed multiplied by time. If we travelled at 100 km/h it would take us approximately 0.44 years to reach the Moon, 89.6 years to reach Mars, 1460 years to reach Saturn and 6590 years to reach Pluto.

a Assuming that there are 365 days in a year, calculate the distance (as a basic numeral) between Earth and:i the Moon ii Mars iii Saturn iv Pluto.

b Write your answers to part a correct to 3 significant figures. c Write your answers to part a using scientific notation correct to 3 significant figures. 17 A light-year is the distance that light travels in one year.

Light travels at approximately 300 000 km/s.a i Calculate the number of seconds in a year. ii Write your answer to part i using scientific

notation.b Calculate the distance travelled by light in one year.

Express your answer: i as a basic numeral ii using scientific notation.c The closest ‘star’ to Earth (other than our sun) is the

star system Alpha Centauri, which is 4.3 light-years away.

i How far is this in kilometres, correct to 4 significant figures? ii Travelling at 100 km/h, how long in years would it take to reach Alpha Centauri?

Alpha Centauri

Exercise 2.5 Scientific notation IndIvIdual PaTHWaYS

⬛ PracTISeQuestions:1–7, 9, 11, 13–15

⬛ cOnSOlIdaTeQuestions:1–8, 10, 12–15, 18

⬛ maSTerQuestions:1–19

FluencY

1 WE17 Write these numbers in decimal notation.a 6.14 × 102 b 6.14 × 103 c 6.14 × 104 d 3.518 × 102

e 1 × 109 f 3.926 73 × 102 g 5.911 × 102 h 5.1 × 103

i 7.34 × 105 j 7.1414 × 106 k 3.51 × 10 l 8.05 × 104

2 WE18a Write these numbers in scientific notation.

a 5000 b 431 c 38 d 350 000

e 72.5 f 725 g 7250 h 725 000 000

3 WE18b Write these numbers in scientific notation, correct to 4 significant figures.

a 43.792 b 5317 c 258.95 d 110.11

e 1 632 000 f 1 million g 123 456 789 h 249.9

4 WE19 Write these numbers in decimal notation.

a 2 × 10−1 b 4 × 10−3 c 7 × 10−4 d 3 × 10−2

e 8.273 × 10−2 f 7.295 × 10−2 g 2.9142 × 10−3 h 3.753 × 10−5

i 5.29 × 10−4 j 3.3333 × 10−5 k 2.625 × 10−9 l 1.273 × 10−15

5 WE20 Write these numbers in scientific notation.

a 0.7 b 0.005 c 0.000 000 3 d 0.000 000 000 01

e 0.231 f 0.003 62 g 0.000 731 h 0.063

6 Write these numbers in scientific notation, correct to 3 significant figures.

a 0.006 731 b 0.142 57 c 0.000 068 3 d 0.000 000 005 12

e 0.0509 f 0.012 46 g 0.000 731 h 0.063

underSTandIng

7 Write these numbers in ascending order.

a 8.31 × 102, 3.27 × 103, 9.718 × 102, 5.27 × 102

b 7.95 × 102, 4.09 × 102, 7.943 × 102, 4.37 × 102

c 5.31 × 10−2, 9.29 × 10−3, 5.251 × 10−2, 5.27 × 10−1

d 8.31 × 102, 3.27 × 103, 7.13 × 10−2, 2.7 × 10−3

8 One carbon atom weighs 1.994 × 10−23 g.

a Write the weight as a decimal.

b How much will 1 million carbon atoms weigh?

c How many carbon atoms are there in 10 g of carbon? Give your answer correct to 4 significant figures.

reFlecTIOn What is the advantage of converting numbers into scientific notation?


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36 Maths Quest 9

18 Scientists used Earth’s gravitational pull on nearby celestial bodies (such as the Moon) to calculate the mass of the Earth at approximately 5.972 sextillion metric tons. a Write 5.972 sextillion using scientifi c notation.b How many signifi cant fi gures does this number

have?

19 An atom consists of smaller particles called protons, neutrons and electrons. Electrons have a mass of 9.109 381 88 × 10−31 kilograms, correct to 9 signifi cant fi gures.a Write the mass of an electron correct to 5 signifi cant fi gures.b Protons and neutrons are the same size. They are both 1836 times the size of an

electron. Use the mass of an electron (correct to 9 signifi cant fi gures) and your calculator to fi nd the mass of a proton correct to 5 signifi cant fi gures.

c Use the mass of an electron correct to 3 signifi cant fi gures to calculate the mass of a proton correct to 5 signifi cant fi gures.

d Why is it important to work with the original amounts and then round to the specifi ed number of signifi cant fi gures at the end of a calculation?

Proton

Neutron

Electrondoc-10816doc-10816doc-10816


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number and algebra

LanguageLanguageLanguage

integersintegersintegersirrational numbersirrational numbersirrational numbersirrational numbersirrational numbersirrational numbersnatural numbersnatural numbersnatural numbersnumeratornumeratornumerator

radical signradical signradical signrational numbersrational numbersrational numbersrecursrecursrecursrecurring decimalrecurring decimalrecurring decimal

scientifi c notationscientifi c notationscientifi c notationsquare rootsquare rootsquare rootsquare rootsquare rootsquare rootsurdsurdsurd

int-0684int-0684int-0684

int-0698int-0698int-0698

int-3202int-3202int-3202

ONLINE ONLY 2.6 ReviewThe Maths Quest Review is available in a customisable format for students to demonstrate their knowledge of this topic.

The Review contains:• Fluency questions — allowing students to demonstrate the

skills they have developed to effi ciently answer questions using the most appropriate methods

• Problem Solving questions — allowing students to demonstrate their ability to make smart choices, to model and investigate problems, and to communicate solutions effectively.

A summary on the key points covered and a concept map summary of this chapter are also available as digital documents.

Review questionsDownload the Review questions document from the links found in your eBookPLUS.

www.jacplus.com.au

Link to assessON for questions to test your readiness FOr learning, your progress aS you learn and your levels OF achievement.

assessON provides sets of questions for every topic in your course, as well as giving instant feedback and worked solutions to help improve your mathematical skills.

www.assesson.com.au


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number and algebra<InveSTIgaTIOn> FOr rICH TaSK Or <number and algebra> FOr PuZZle

38 Maths Quest 9

InveSTIgaTIOn

Concentric squares

4

3

2

1

X

rICH TaSK

1 Use the diagram to complete the following table, leaving your answers in simplest surd form, if necessary. The fi rst square has been completed for you.

Observe the patterns in the table, and answer questions 2 and 3.2 What would be the side length of the tenth square of this pattern?3 What would be the length of the diagonal of this tenth square?


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number and algebra

Consider, now, a different arrangement of the squares on one-centimetre grid paper, as shown in the diagram.

These squares all still have a central point, labelled Y.4 Use the diagram to complete the following table for these squares, leaving your answers in simplest surd

form if necessary.

5 What would be the side length of the tenth square in this pattern of squares?

6 What would be the length of the diagonal of the tenth square?

Use the two diagrams above to answer questions 7 to 11.7 In which arrangement does the square have the greater side length?8 Which arrangement shows the squares with the greater diagonal length?9 Compare the area of a square in the fi rst diagram with the area of the corresponding square in the second

diagram.10 Compare the perimeter of corresponding squares in the two diagrams.11 Examine the increase in area from one square to the next in the set. What increase in area would you

expect from square 7 to square 8 in each of the patterns?

Consider a set of concentric circles around a centre Z, drawn on one-centimetre grid paper.12 Investigate the change in circumference of the

circles, moving from one circle to the next (smallest to largest). Write a general formula for this increase in circumference in terms of π.

13 Write a general formula in terms of π and r that can be used to calculate the increase in area from one circle to the next (smallest to largest).

4

3

2

1

Y

4

3

2

1

Z


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number and algebra<InveSTIgaTIOn> FOr rICH TaSK Or <number and algebra> FOr PuZZle

40 Maths Quest 9

number and algebra

COde PuZZle

The answer to each question and the letter besideit give the puzzle’s answer code.

A = 3 5 + 5 =

H = 3 6 – 2 6 =

L = 7 5 – 5 5 =

E = 10 3 – 4 3 =

D = 18 – 2 2 =

M = 7 6 – 54 =

N = 45 – 20 =

O = 6 7 – 28 =

A = 5 2 + 3 3 – 2 =

B = 108 – 5 3 =

B = 5 2 + 3 2 =

S = 5 + 3 5 – 3 =

D = 2 + 2 5 + 3 2 – 5 =

C = 3 + 3 =

T = 8 + 18 – 2 =

E = 200 – 147 =

D = 2 6 + 3 6 =

U = 12 – 32 + 6 2 =

H = 5 – 2 2 + 9 =

E = 2 7 + 7 =

E = 50 + 27 – 5 2 =

O = 75 + 4 5 + 12 =

A = 8 + 3 2 =

3 6 2 3 4 2 + 3 3

5 3 4 2

2 2 + 2 3 4 5 – 3 3 7 7 3 + 4 5 5 6 3

W = 3 + 4 + 5 =

S = 8 5 – 45 – 20 =

F = 3 3 + 12 =

A = 48 – 2 3 + 20 =

E = 150 + 2 6 – 96 =

4 7

8 2 2 5 5 2 5 6 3 32 3 + 2 5 4 2 + 5 10 2 – 7 3

2 7 4 6 4 5 3 58 – 2 2 2 + 3 + 5

3

6 2

E = 6 7 – 4 7 =

When 3 people fell in the water, why did only 2 of them get their hair wet?


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Activities


2.1 Overviewvideo • The story of mathematics: The evolution of

numbers (eles-1689)

2.2 rational numbersdigital doc • SkillSHEET (doc-6100): Operations with directed

numbers

2.3 Surdsinteractivity• Balancing surds (int-2762)digital docs• SkillSHEET (doc-6101): Calculating the square

root of a number• SkillSHEET (doc-6102): Using a calculator to

evaluate numbers in index form

• SkillSHEET (doc-10813): Simplifying surds • WorkSHEET 2.1 (doc-6119): Surds

2.4 real numbersdigital doc• SkillSHEET (doc-10814): Rounding to a given

number of signifi cant fi gures

2.5 Scientifi c notationdigital docs• SkillSHEET (doc-10815): Multiplying and dividing

by powers of 10 • WorkSHEET 2.2 (doc-10816): Scientifi c notation

2.6 reviewInteractivities • Word search (int-0684)• Crossword (int-0698)• Sudoku (int-3202)

To access ebookPluS activities, log on to www.jacplus.com.au

number and algebra


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42 Maths Quest 9

Exercise 2.2 — Rational numbers 1 a 15

1 b −8

1 c 8

3 d −41

8

e 41 f 73

10 g 2

1000 h 872

10

i 01 j 156

100 k 3612

1000 l −8

100

2 a 0.555 555 55 b 0.515 151 51 c 0.511 111 11 d 6.031 313 1 e 5.183 183 1 f −7.024 444 4 g 8.912 491 2 h 5.123 434 3 i 5.123 412 3 j 3.002 020 2 3 a 0.5

# b 0.2

#7# c 2.1

#8# d 0.4

#28 571

#

e −1.7#4# f 0.7

#3# g −3.83

# h 0.46

#

i 0.4#6# j 0.04

#6#

4 a 15 b −3, 0, 15 c −3, −37

, 0, 2.3, 2.3#, 23

5, 15

5 2.1#, 2.1

#21

#, 2.12

#1#, 2.1

#2#, 2.12

#

6 Every integer a can be written as a1

and is therefore rational.

7 Yes, because the division of an integer by another integer will always result in a rational number.

8 a 13, 2

3, 1

6 b 6

9 Answers will vary. Example answers are shown.

a 19 b 10

99 c 100

999 d 1000

9999

10 $19 990.20 11 657

Exercise 2.3 — Surds 1 a 1 b 2 c 0 d 1

3

e 54 f 0.4 g 20 h 100

i 25 j 1.2 k 4.5 l 1000

m 0.03 n 16 2 a 9 b −9 c 11 d −21 e 2 f 4 g 7 h 3 i 4 j 5 k −7 l −3 3 b, d, g, h, k, m, o 4 a !21 b −!21 c 2!6 d 6!7 e 10!14 f 15!10 g 12!7 h 9!7 i 22!10 j 22!3 k 3 l 24 m 6 n 11 o 51 p 30 5 a !3 b −!2 c !6 d !5

e 3!3 f 3!6

2 g 15!2 h

5!53

i −2!5 j !3 k !3 l !7 m !35 n 2 o −3 p 3 6 a 2!5 b 2!2 c 3!2 d 7 e !30 f 5!2 g 2!7 h 6!3 i 12!2 j 4!3 k 10!5 l 9!2 7 a 4!2 b 15!3 c 48 d 35!2 e 20!6 f 10!3 g 4!42 h 72!2 i 27!5 j 132!2 8 a !12 b !175 c !108 d !80 e !384 f !90 g !32 h !720 i !600 j !338 9 a D b A c C d C e D f D g D h B 10 a 15!3 b 19!5 c 16!7 d 17!2 e a!c f d

2!b g hk!j h f!f

11 a 2!2 b −4!5 c −6!3 d 0 e 5!11 f 2!7 g 10!2 + 7!3 h 8!5 + !6 i 8!10 + 7!3 j 16!2 − 11!5 k 2!6 l 5!5 + 15 12 a !2 b 0 c −5!3 d −4!7 e 5!6 + 6!5 f 2!3 − 3!5 g 4!6 + 6!5 − 14!2 h 29!5 + 22!3 i 9!11 − !30 j 28!2 − 39!5 13 a D b A c C d A 14 a 2!3 b 5!2 c 10!2 d 2!10 15 a 2!7, 4!2, 3!6, !60, 8, 6!2 b !2, !3, 2, !6, 2!2, 3, 2!3 16 a !2 m b 1.41 m

c i Any of these:

ii 14!2 m iii 19.80 m

17 "a3b2 = "a3 × "b2

= "a2 × "a × "b2

= a × "a × b

= ab"a 18 36!5 m 19 (1750!2 + 2750!3 + 320) m 20 The side length will be rational if x is a perfect square; otherwise,

the side length will be a surd.21 a i !13 ii 3 iii !38

b For side lengths !a and !b, the length of the hypotenuse is !a + b.

c i !1500 = 10!15 ii !456 = 2!144 iii !187

d The answer should be !b + a = !c.

Challenge 2.1452 = 2025; 562 = 3136

Exercise 2.4 — Real numbers 1 a 1.571 b 2.236 c 3.873 d 16.435 e 5.111 f 5.156 g 5.152 h 2.718 i 137.358 j 0.429 k 0.077 l 2.429

AnswersTOPIC 2 Number skills


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number and algebra


m 1.000 n 6.581 o 4.000 p 0.254 q 0.025 r 0.001 s 5.000 t 2342.156 2 3.1416, 3.141 59, 3.141 593, 3.141 592 7 3 a 2 b 3 c 4 d 3 e 4 f 4 g 2 h 1 i 3 j 4 k 2 l 3 4 a 1.5708 b 2.2361 c 3.8730 d 16.435 e 5.1111 f 5.1556 g 5.1515 h 2.7183 i 137.36 j 0.428 57 k 0.076 923 l 2.4286 m 1.0000 n 6.5813 o 4.0000 p 0.253 64 q 0.025 400 r 0.000 913 60 s 5.0000 t 2342.2 5 a !7, 3.5, !15, 4, !18

b 3.14, 3.1#4, π, 22

7, !10

6 a F b T c F d T 7 a F b T c F d T 8 a T b F c F d T 9 !25 means the positive square root of 25, i.e. 5, because

52 = 25. Similarly, −!25 means the negative square root of 25, which is −5 because (−5)2 = 25. But −25, like all other negative numbers, has no real square root, so !−25 is ‘undefined’.

10 The 5th digit (5) causes the 4th digit to round up from 9 to 10, which causes the 3rd digit to round up from 9 to 10, which causes the 2nd digit to round up from 2 to 3.

11 a 153.86 cm2 b 153.9380 cm2 c 153.94 cm2

d Yes, because 3.14 is used as an estimate and is not the exact value of π as used on the calculator.

12 a 2617.9939 cm3 b 8 c Yes, any number with a finite number of decimal places is a

rational number. 13 13 000Exercise 2.5 — Scientific notation 1 a 614 b 6140 c 61 400 d 351.8 e 1 000 000 000 f 392.673 g 591.1 h 5100 i 734 000 j 7 141 400 k 35.1 l 80 500 2 a 5.00 × 103 b 4.31 × 102 c 3.8 × 101

d 3.5 × 105 e 7.25 × 101 f 7.25 × 102

g 7.25 × 103 h 7.25 × 108

3 a 4.379 × 101 b 5.317 × 103 c 2.590 × 102

d 1.101 × 102 e 1.632 × 106 f 1.000 × 106

g 1.235 × 108 h 2.499 × 102

4 a 0.2 b 0.004 c 0.0007 d 0.03 e 0.082 73 f 0.072 95 g 0.002 914 2 h 0.000 037 53 i 0.000 529 j 0.000 033 33 k 0.000 000 002 625 l 0.000 000 000 000 001 273 5 a 7 × 10−1 b 5 × 10−3 c 3 × 10−7 d 1 × 10−11 e 2.31 × 10−1 f 3.62 × 10−3

g 7.31 × 10−4 h 6.3 × 10−2

6 a 6.73 × 10−3 b 1.43 × 10−1 c 6.83 × 10−5

d 5.12 × 10−9 e 5.09 × 10−2 f 1.25 × 10−2

g 7.31 × 10−4 h 6.30 × 10−2

7 a 5.27 × 102, 8.31 × 102, 9.718 × 102, 3.27 × 103

b 4.09 × 102, 4.37 × 102, 7.943 × 102, 7.95 × 102

c 9.29 × 10−3, 5.251 × 10−2, 5.31 × 10−2, 5.27 × 10−1

d 2.7 × 10−3, 7.13 × 10−2, 8.31 × 102, 3.27 × 103

8 a 0.000 000 000 000 000 000 000 019 94 g b 1.994 × 10−17 g c 5.015 × 1023 atoms 9 3844 hours ≈ 160.17 days 10 333 333.33

11 The neutron is heavier by 2.3 × 10−30 kg. 12 a 2.1 × 107 b 1.29 × 108

13 a Check with your teacher. b i 5.36 × 107 ii 5.36 × 107

c The answers are the same because 2.95 × 103 is very small compared with 5.36 × 107.

14 8.75 × 10−7 is such a small amount (0.000 000 875) that when it is added to 2.39 × 10−3, it doesn’t affect the value when given to 3 significant figures.

15 1.5 × 10−2, 14%, 0.52, 58, 0.753, !3 2, !2

16 a i 385 440 km ii 78 489 600 km iii 1 278 960 000 km iv 5 772 840 000 km b i 385 000 km ii 78 500 000 km iii 1 280 000 000 km iv 5 770 000 000 km c i 3.85 × 105 km ii 7.85 × 107 km iii 1.28 × 109 km iv 5.77 × 109 km 17 a i 31 536 000 ii 3.1536 × 107

b i 9 460 800 000 000 ii 9.4608 × 1012

c i 4.068 × 1013 ii 12 900 years 18 a 5.972 × 1021 b 4 19 a 9.1094 × 10−31 b 1.6725 × 10−27

c 1.6726 × 10−27

d It is important to work with the original amounts and leave rounding until the end of a calculation so that the answer is accurate.

Challenge 2.21.394 × 106 km; 12 800 or 1.28 × 104 km

Investigation — Rich task 1 Square 1 2 3 4

Side length (cm) 2 4 6 8

Diagonal length (cm) 2!2 4!2 6!2 8!2

Perimeter (cm) 8 16 24 32

Area (cm2) 4 16 36 64

2 20 cm 3 20!2 cm 4 Square 1 2 3 4

Side length (cm) !2 2!2 3!2 4!2

Diagonal length (cm) 2 4 6 8

Perimeter (cm) 4!2 8!2 12!2 16!2

Area (cm2) 2 8 18 32

5 10!2 cm 6 20 cm 7 First diagram 8 First diagram 9 The squares in the first diagram are twice the area of the

corresponding square in second diagram. 10 The squares in the first diagram have a perimeter that is !2 times

the perimeter of the corresponding square in the second diagram. 11 An increase of 60 cm2 in the first diagram and an increase of

30 cm2 for the second diagram 12 2π 13 π(2r + 1)

Code puzzleBecause one of them was bald-headed.


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